KosterlitzThoulessHalperinNelsonYoung

The Kosterlitz-Thouless transition (KT transition) is a topological phase transition in two-dimensional systems with a continuous symmetry, notably in the two-dimensional XY model. It arises from the binding and unbinding of topological defects—vortex-antivortex pairs. The theory was developed by Vadim Berezinskii, and by John Kosterlitz and David Thouless in 1973.

In two dimensions, true long-range order is forbidden at finite temperature by the Mermin-Wagner theorem, but quasi-long-range order can exist. Below the transition temperature Tc, vortices exist only as bound pairs, leading to algebraic decay of correlations. Above Tc, free vortices proliferate and destroy quasi-long-range order, producing exponential decay of correlations. The transition is of infinite order with essential singularities; the correlation length diverges as ξ ∼ exp(b/√(T−Tc)) when approaching Tc from above. There is no conventional local order parameter.

A hallmark of the KT transition is the universal jump in the superfluid stiffness (or spin stiffness) at Tc, as predicted by Nelson and Kosterlitz. There is no spontaneous symmetry-breaking order parameter; the transition is topological, governed by vortex excitations rather than a local order parameter. Renormalization-group analyses and finite-size scaling provide the theoretical framework for understanding the transition.

KT transitions are observed in thin helium films, two-dimensional superconductors, Josephson junction arrays, and ultracold atomic gases confined to two dimensions. They have become a canonical example of topological phase transitions in low-dimensional systems.